Geometry: Using and Proving Angle Supplements
Using and Proving Angle Supplements
When acute angles need a supplement, they turn to an obtuse angle. Even though I am an authority on the subject, and I make that statement with a certain level of confidence and conviction, skeptics should demand that I put my money where my mouth is and prove it. And to set a good example, I will.
- Example 6: Prove that the supplement of an acute angle is an obtuse angle.
- Solution: You know what to do, so just do it.
- 1. State the theorem.
- Theorem 9.5: The supplement of an acute angle is an obtuse angle.
- 2. Draw a picture. Figure 9.6 shows an acute angle ∠ABC and its supplement ∠CBD. Together ∠ABC and ∠CBD form the straight angle ∠ABD.
- 3. Interpret what you are given in terms of your drawing. You are given ∠ABC and its supplement ∠CBD, with ∠ABC acute.
- 4. Interpret what you are trying to prove in terms of your drawing. Prove that ∠CBD is obtuse.
- 5. Prove the theorem. What's the game plan? This proof involves acute, obtuse, and supplementary angles, so you'll probably use their definitions somewhere. Because you'll be dealing with inequalities (acute angles have measure less than 90º and obtuse angles have measure greater than 90º), you might need your definitions of < or >, and you might need our Protractor Postulate. And there's always algebra.
|1.||∠ABC is acute, and ∠ABC and ∠CBD are supplementary.||Given|
|2.||m∠ABC + m∠CBD = 180º||Definition of supplementary angles|
|3.||m∠ABC < 90 º||Definition of acute angle|
|4.||m∠ABC + 90º < 180º||Addition property of inequality|
|5.||m∠ABC + 90º < m∠ABC + m∠CBD||Substitution (steps 2 and 4)|
|6.||90º < m∠CBD||Subtraction property of inequality|
|7.||∠CBD is obtuse||Definition of obtuse angle|
Now that you're starting to crank out those formal proofs, it's time to open things up and see how you perform on the open road. Take a look at Figure 9.7. ∠ABC and ∠CBD are adjacent supplementary angles. If you construct the bisectors of each of these two angles, then together the bisectors will form a new angle. But not just any angle. This new angle will be right. And you'll prove it.
- Example 7: Prove that the bisectors of two adjacent supplementary angles form a right angle.
- Solution: Follow these steps.
- 1. State the theorem.
- Theorem 9.6: The bisectors of two adjacent supplementary angles form a right angle.
- 2. Draw a picture (see Figure 9.7).
- 3. Interpret the given information in terms of the picture. ∠ABC and ∠CBD are adjacent supplementary angles; →BE bisects ∠ABC, and →BF bisects ∠CBD.
- 4. Interpret what to prove in terms of the picture. Prove that ∠EBF is a right angle.
- 5. Prove the theorem. Your game plan: You'll need some definitions in this proof: supplementary angles, right angles, and angle bisectors. Because you will be breaking up angles, the Angle Addition Postulate might be useful. Let's see how it all unfolds.
|1.||∠ABC and ∠CBD are adjacent supplementary angles; →BE bisects ∠ABC, and →BF bisects ∠CBD||Given|
|2.||∠ABE ~= ∠EBC, ∠CBF ~= ∠FBD||Definition of angle bisector|
|3.||m∠ABE = m∠EBC, m∠CBF = m∠FBD||Definition of ~=|
|4.||m∠ABC + m∠CBD = 180º||Definition of supplementary angles|
|5.||m∠ABE + m∠EBC = m∠ABC, m∠CBF + m∠FBD = m∠CBD, and m∠EBC + m∠CBF = m∠EBF||Angle Addition Postulate|
|6.||m∠ABE + m∠EBC + m∠CBF + m∠FBD = 180º||Substitution (steps 4 and 5)|
|7.||2m∠EBC + 2m∠CBF = 180º||Substitution (steps 3 and 6)|
|8.||m∠EBC + m∠CBF = 90º||Algebra|
|9.||m∠EBF = 90º||Substitution (steps 5 and 8)|
|10.||∠EBF is right||Definition of right angle|
Keep in mind that there is more than one way to construct a proof. If you put three mathematicians in a room and have them prove the same theorem, you will probably get three different proofs. They would all be valid (assuming they did it right), though they might have taken different steps along the way. Variety is the spice of life. Just be sure to avoid using cheap reasons in your proofs. Trust me: It will show.
Put Me in, Coach!
Here's your chance to shine. Remember that I am with you in spirit and have provided the answers to these questions in Answer Key.
- 1. If E is between D and F, write a formal proof that DE = DF − EF.
- 2. Given ∠ABC and →BD as in Figure 9.8, write a formal proof that m∠ABD = m∠ABC − m∠DBC.
- 3. Prove that the angle bisector of an angle is unique.
- 4. Prove that the supplement of a right angle is a right angle.
Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.